Vol. 10 No 3 - Pi Mu Epsilon

ON THE GIRTHS OF REGULAR PLANAR GRAPHSMasakazu NihtvFujishiro High SchoolOne of the most fascinating yet mysterious classes of graphs are the cages.We introduce a planar version, classify them. and use this information topresent another proof of the fact that there are exactly five Platinic solids.We begin with a few definitions. The degree of a vertex v in a graph G isthe number of edges of G incident with v. A graph in which every vertex hasthe same degree is called a regular graph; if every vertex has degree k, thegraph is called a k-regular graph. The cardinality of the vertex set of G iscalled the order of G and is denoted by p. while the cardinality of its edge setis the size of G and is denoted by q. The length of the shortest cycle in a graphG that contains cycles is called the girth of G and is denoted by g(G) or g.Let us consider the k-regular graphs with girth g. The minimal order of ak-regular graph with girth g is denoted by f (k, g ) , and the k-regular graphs ofgirth g and order f(k, g) are called (k, g)-cages . For example, f ( 3,4) = 6and f (3,5) = 10. The ( 3,4)-cage and (3, 5) -cage are unique and nonplanar,[I, 236-2391, 12.34-431. They are shown below.Fig. 1: (3,4) -cageFig. 2: @,5) -cageis not true.We will first determine the girths of all planar k-regular graphs for k = 4,5.THEOREM 1. If G is a connected planar +regular or 5-regular graph, theng(G) = 3.Proof. Let p, q, and s denote the order, size, and number of faces .of G.Then we have(1)andby Euler's formula.Let the distinct lengths of the boundaries of the faces of G be denoted byg = go,g,,-,g", (go sg,, i= L2,-. m).Suppose that there are s, faces with boundary of length g, . Then we have(3) E^=o s,g = 29.From (1) and (2) we have(4) y,":o s, = 2 +q(k- 2)/k,and from (3) we also have(5) O